Remark 2.26 We note that the above proof is very different to the result which appears in the abstract work of [1]}, however the result still shows a moving space equivalence.
In the abstract setting of [1]}, the transformations satisfy the condition that \(T_t := \phi _t^{*}\phi _t\)  takes \(H_0\)  to \(H_0\) .
This is not the case with the definition of \(\phi _t^*\)  which appears in this work.
The transformations considered by [3]} which are applications of the theory of [1]} are 'orthogonal' in the sense that \(\phi _t^* = \phi _{-t}\) .
In this setting, one could potentially change the definition of \(\phi _t^*\)  to achieve \(\phi _t^*\phi _t\colon H_0 \rightarrow H_0\) , however one may then lose the ability to write down a meaningful interpretation of \(\phi _t^*\) .
